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Properties

Label 2-525-7.2-c1-0-18
Degree $2$
Conductor $525$
Sign $0.386 + 0.922i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (0.5 + 0.866i)3-s + (−0.999 − 1.73i)4-s + 1.99·6-s + (2.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)11-s + (1 − 1.73i)12-s − 13-s + (1.00 − 5.19i)14-s + (1.99 − 3.46i)16-s + (0.999 + 1.73i)18-s + (−0.5 + 0.866i)19-s + (2 + 1.73i)21-s + 3.99·22-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)2-s + (0.288 + 0.499i)3-s + (−0.499 − 0.866i)4-s + 0.816·6-s + (0.944 − 0.327i)7-s + (−0.166 + 0.288i)9-s + (0.301 + 0.522i)11-s + (0.288 − 0.499i)12-s − 0.277·13-s + (0.267 − 1.38i)14-s + (0.499 − 0.866i)16-s + (0.235 + 0.408i)18-s + (−0.114 + 0.198i)19-s + (0.436 + 0.377i)21-s + 0.852·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10238 - 1.39846i\)
\(L(\frac12)\) \(\approx\) \(2.10238 - 1.39846i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-2.5 + 0.866i)T \)
good2 \( 1 + (-1 + 1.73i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (4.5 + 7.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (1.5 + 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (8 - 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.73802854835057676488492320423, −10.17070283025389549463381514989, −9.234491149321724828970440152883, −8.087545820264517661110708125727, −7.19130219341138338063445696667, −5.55188558746171991350889999522, −4.55224518815141759075735586012, −3.98643922105017251628218171753, −2.69916768281988765202068977913, −1.57545241332423703251274765201, 1.71715275112797529419618733811, 3.41242731166027719017887927526, 4.75744198366204605797690189069, 5.44759545558681018086493052816, 6.55712524828348869494638242400, 7.18081476956212998108015151422, 8.311258406073336097362721566038, 8.600700266355208737024112616735, 10.10883393176178781629677559448, 11.23278865843254003345597919929

Graph of the $Z$-function along the critical line